I think I now know that these questions asked for (after studing CFA level 2 in these few days...). May not be the correct proof but I think the idea will be as follows:
1. Prove that if R^2 = 1, then the estimated slope of the regression of y on x
equals to the reciprocal of the estimated slope of the regression of x on y.
If R^2 = 1, x and y are prefectly correlated,
i.e. for linear relationship y= a+bx+E , error term E = 0 for all x.
=> y = a+bx, slope = b
=> x = y/b - a/b, slope = 1/b, which is reciprocal of b.
2. Prove that the value of R^2 is unchanged if we change the units in both the
x and y–coordinates via x′i = c1 + c2xi and y′i = d1 + d2yi for i = 1, . . . , n.
The main idea is that changing x and y in a linear manner, the relationship between x and y are still linear. R^2 should be no effect.
e.g. y = ax+b, if let x′i = c1 + c2xi and y′i = d1 + d2yi => (x′i - c1) / c2 = xi , (y′i - d1) / d2 = yi,
substititon gets (y′i - d1) / d2 = a(x′i - c1) / c2 + b
=> y′i = ad2(x′i - c1) / c2 + bd2 + d1
=> y′i = ad2 / c2(x′i ) + (bd2 + d1 - ac1d2 / c2), also linear.
